It is known that for a discrete memory-less channel with input-output relation as
\(Y_i = X_i + N_i\) , where the noise complex-valued samples \(N_i \sim \mathcal {CN}(0,\sigma ^2)\)  are i.i.d. and independent of \(X_i\) , and with a power constraint on the input signal \(\mathbb {E}[|X_i|^2] \le P\) , the mutual information \(I(X_i;Y_i)\)  is maximized when \(X_i \sim \mathcal {CN}(0,P)\)  and it corresponds to the channel capacity \(C = \log _2(1+\text{SNR})\) , where SNR denotes the signal-to-noise power ratio.
When dealing with a pair of input-output \(d\) -dimensional Gaussians \(X\)  and \(Y\) , each with i.i.d. zero-mean components, the capacity can be written in terms of correlation coefficient [1]} \(\rho = \mathbb {E}[X_i \cdot Y_i]/ \sqrt{\mathbb {E}[X_i^2]\cdot \mathbb {E}[Y_i^2]}\) , for \(i=1,\dots ,d\) , as
\(I(X;Y) = -\frac{d}{2} \log _2(1-\rho ^2).\) 
